It depends on what you want. Electric guitarists rarely use capos because open strings are less important. If you want ringing open strings in a different key, which tuning you use depends I the key, but tuning between each song means you better have many instruments or great between-song patter.
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VarLogRantFeb 6 '13 at 12:02

2 Answers
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A capo is for transposing a piece up by some number of semitones, without changing the fingering. So let's look at ways to transpose those pieces without a capo -- it will often involve changing the fingering:

If the piece does not contain any open strings, you don't need a capo. Just play the piece with your left hand higher up the fretboard.

If the piece contains a few open strings, you may be able to finger the piece a different way -- whenever the original had an open string, you'll need to fret it. Often you can achieve this with a barre. In this case your index finger is being a "capo" temporarily.

You could tune all your strings up by whatever interval you're transposing. But note that tuning up by more than 3 semitones puts you at risk of breaking strings (and when this happens, cutting flesh), or damaging your instrument's neck.

You could accept different chord voicings. For example, if your original involved strumming the chords D then G, and you're transposing up by 2 semitones, then you could play the chords E then A, including open strings. It will sound different, but it would still be an acceptable accompaniment to a song.

There are transpositions that can't practically be achieved without a capo.

Consider this chord:

%0.7/2.7/3.6/1.0.0

The fretting is all on the 6th and 7th fret, but there are also three open strings. If you tried to transpose this up by 3 semitones, without a capo or retuning, you'd have to finger the 3rd fret, as well as the 9th and 11th fret. Nobody's hand is that big!

You don't explain why you don't want to use a capo. If you want to achieve what a capo does, then a capo is the best way to do it.

For the sake of a thorough answer, there are some new electric guitars that have recently appeared on the market that make use of what is refered to as a "digital capo". They can be configured to use built-in digital signal processing to raise or lower the pitch of the strings by digital pitch-shifting, while the actual tuning of the physical strings stays the same. So the notes coming out of your amplifier are different than the acoustical notes played by the strings themselves.

Getting one of these guitars would be expensive as they are rather esoteric instruments. The pitch-shifting technique is not entirely natural-sounding.

I understand this is how Joni Mitchell plays these days, using the box to switch between the outstanding number of tunings she uses, although then it would be "alternative to alternate tunings" instead.
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VarLogRantFeb 10 '13 at 1:07

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These instruments, at least the Variax and the Fender-Roland, are set up to provide all sorts of alternate tunings, but you can also used them to raise or lower the pitch of all six strings by the same amount at once, acting as a capo.
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Wheat WilliamsFeb 10 '13 at 3:53